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Given a quadratic form Q(x,y)=x^2+y^2, (1) then Q(x,y)Q(x^',y^')=Q(xx^'-yy^',x^'y+xy^'), (2) since (x^2+y^2)(x^('2)+y^('2)) = (xx^'-yy^')^2+(xy^'+x^'y)^2 (3) = ...

A topological space.

A diffeomorphism is a map between manifolds which is differentiable and has a differentiable inverse.

A pair of overdots placed over a symbol, as in x^.., most commonly used to denote a second derivative with respect to time, i.e., x^..=d^2x/dt^2.

A space endowed with a non-Euclidean elliptic geometry.

The difference between a quantity and its estimated or measured quantity.

A connection between two or more vertices of a hypergraph. A hyperedge connecting just two vertices is simply a usual graph edge.

Two quantities y and x are said to be inversely proportional (or "in inverse proportion") if y is given by a constant multiple of 1/x, i.e., y=c/x for c a constant. This ...

A square matrix which is not reducible is said to be irreducible.

Topological lower bounds in terms of Betti numbers for the number of critical points form a smooth function on a smooth manifold.